| L(s) = 1 | + 0.599·2-s − 2.83·3-s − 1.64·4-s − 1.70·6-s − 0.360·7-s − 2.18·8-s + 5.04·9-s − 0.625·11-s + 4.65·12-s − 4.68·13-s − 0.216·14-s + 1.97·16-s + 8.17·17-s + 3.02·18-s − 4.27·19-s + 1.02·21-s − 0.374·22-s + 2.92·23-s + 6.18·24-s − 2.80·26-s − 5.79·27-s + 0.591·28-s − 9.18·29-s − 2.11·31-s + 5.54·32-s + 1.77·33-s + 4.89·34-s + ⋯ |
| L(s) = 1 | + 0.423·2-s − 1.63·3-s − 0.820·4-s − 0.694·6-s − 0.136·7-s − 0.771·8-s + 1.68·9-s − 0.188·11-s + 1.34·12-s − 1.29·13-s − 0.0577·14-s + 0.493·16-s + 1.98·17-s + 0.712·18-s − 0.979·19-s + 0.223·21-s − 0.0799·22-s + 0.610·23-s + 1.26·24-s − 0.550·26-s − 1.11·27-s + 0.111·28-s − 1.70·29-s − 0.379·31-s + 0.980·32-s + 0.308·33-s + 0.840·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4027222065\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4027222065\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 - 0.599T + 2T^{2} \) |
| 3 | \( 1 + 2.83T + 3T^{2} \) |
| 7 | \( 1 + 0.360T + 7T^{2} \) |
| 11 | \( 1 + 0.625T + 11T^{2} \) |
| 13 | \( 1 + 4.68T + 13T^{2} \) |
| 17 | \( 1 - 8.17T + 17T^{2} \) |
| 19 | \( 1 + 4.27T + 19T^{2} \) |
| 23 | \( 1 - 2.92T + 23T^{2} \) |
| 29 | \( 1 + 9.18T + 29T^{2} \) |
| 31 | \( 1 + 2.11T + 31T^{2} \) |
| 37 | \( 1 + 2.51T + 37T^{2} \) |
| 41 | \( 1 + 2.70T + 41T^{2} \) |
| 43 | \( 1 + 1.59T + 43T^{2} \) |
| 47 | \( 1 - 5.73T + 47T^{2} \) |
| 53 | \( 1 + 9.95T + 53T^{2} \) |
| 59 | \( 1 + 6.89T + 59T^{2} \) |
| 61 | \( 1 - 4.62T + 61T^{2} \) |
| 67 | \( 1 + 6.13T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 - 0.559T + 73T^{2} \) |
| 79 | \( 1 - 4.10T + 79T^{2} \) |
| 83 | \( 1 - 15.1T + 83T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.80457076608658301964661013088, −7.32456310265937137062575771813, −6.37108093304335517086371486360, −5.73961911382699648494237841119, −5.19263345564429235467610545067, −4.77515806593306320426862696470, −3.87831153385195818464600037012, −3.03026038519022950748111556940, −1.56612676531321506479381602185, −0.34982431089390506999981654639,
0.34982431089390506999981654639, 1.56612676531321506479381602185, 3.03026038519022950748111556940, 3.87831153385195818464600037012, 4.77515806593306320426862696470, 5.19263345564429235467610545067, 5.73961911382699648494237841119, 6.37108093304335517086371486360, 7.32456310265937137062575771813, 7.80457076608658301964661013088
